For stationary ergodic processes the set of calibration-passing forecast distributions equals the mean-preserving contractions of the conditional distribution, allowing the dynamic game to be solved via static persuasion.
The Annals of Mathematical Statistics 24, 2 (1953), 265–272
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Calibrated Forecasting and Persuasion
For stationary ergodic processes the set of calibration-passing forecast distributions equals the mean-preserving contractions of the conditional distribution, allowing the dynamic game to be solved via static persuasion.